Occasionally there are exceptional voices having a wider range than the above scale indicates.

Ratio of Intervals:—

Octave1— 2
Perfect fifth2— 3
Perfect fourth3— 4
Major third4— 5
Minor third5— 6
Major sixth3— 5
Minor sixth5— 8
Major second8— 9
Minor second15—16
Major seventh8—15
Minor seventh9—16

Each tone generates "over tones" called harmonics. These harmonics are the octave, the twelfth (perfect fifth), the seventeenth (major third), the twenty-first (minor seventh) and the twenty-third (minor ninth). Other harmonics than the above exist but are not used at the present time in chord construction. The old theorists treated chords of the eleventh and thirteenth, but modern theorists treat these intervals as suspensions, anticipations, etc.[D] The origin of chord construction may be seen from these harmonics. These over tones, generated from a fundamental, are the pure (untempered) intervals. The tempered intervals, with the exception of the octave, are slightly out of tune but not enough so to shock the ear.

The pure (untempered) scale of C has the following number of vibrations per second:—

Middle c 261.0
d 293.6
e 326.2
f 348.0
g 391.5
a 435.0
b 489.3

An entire volume would be necessary to explain completely the science of acoustics. All ambitious students should consult books on acoustics. The author recommends the books on sound by the following writers:—